## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 42, Number 2 (2002), 207-221.

### Nearly holomorphic functions and relative discrete series of weighted $L^2$-spaces on bounded symmetric domains

#### Abstract

Let $\Omega = G/K$ be a bounded symmetric domain in a complex vector space $V$ with the Lebesgue measure $dm(z)$ and the Bergman reproducing kernel $h(z,w)^{-p}$. Let $d\mu _{\alpha}(z) = h(z, \bar{z})^{\alpha}dm(z)$, $\alpha > -1$, be the weighted measure on $\Omega$. The group $G$ acts unitarily on the space $L^{2}(\Omega , \mu_\alpha )$ via change of variables together with a multiplier. We consider the discrete parts, also called the relative discrete series, in the irreducible decomposition of the $L^{2}$-space. Let $\bar{D} = B(z, \bar{z})\partial$ be the invariant Cauchy-Riemann operator. We realize the relative discrete series as the kernels of the power $\bar{D}^{m+1}$ of the invariant Cauchy-Riemann operator $\bar{D}$ and thus as nearly holomorphic functions in the sense of Shimura. We prove that, roughly speaking, the operators $\bar{D}^{m}$ are intertwining operators from the relative discrete series into the standard modules of holomorphic discrete series (as Bergman spaces of vector-valued holomorphic functions on $\Omega$).

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 42, Number 2 (2002), 207-221.

**Dates**

First available in Project Euclid: 14 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250283866

**Digital Object Identifier**

doi:10.1215/kjm/1250283866

**Mathematical Reviews number (MathSciNet)**

MR1966833

**Zentralblatt MATH identifier**

1028.43012

**Subjects**

Primary: 43A85: Analysis on homogeneous spaces

Secondary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX] 32M15: Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15]

#### Citation

Zhang, Genkai. Nearly holomorphic functions and relative discrete series of weighted $L^2$-spaces on bounded symmetric domains. J. Math. Kyoto Univ. 42 (2002), no. 2, 207--221. doi:10.1215/kjm/1250283866. https://projecteuclid.org/euclid.kjm/1250283866